Hollow-core fibre and method of manufacturing thereof

ABSTRACT

A hollow-core fibre (100) of non-bandgap type comprises a hollow core region (10) axially extending along the hollow-core fibre (100) and having a smallest transverse core dimension (D), wherein the core region (10) is adapted for guiding a transverse fundamental core mode and transverse higher order core modes, and an inner cladding region (20) comprising an arrangement of anti-resonant elements (AREs) (21, 21A, 21B) surrounding the core region (10) along the hollow-core fibre (100), each having a smallest transverse ARE dimension (di) and being adapted for guiding transverse ARE modes, wherein the core region (10) and the AREs (21, 21A, 21B) are configured to provide phase matching of the higher order core modes and the ARE modes and the ARE dimension (di) and the core dimension (D) are selected such that a ratio of the ARE and core dimensions (di/D) is approximated to a quotient of zeros of Bessel functions of first kind (ulm,ARE/ulm,core), multiplied with a fitting factor in a range of 0.9 to 1.5, with m being the m-th zero of the Bessel function of first kind of order l, said zeros of the Bessel functions describing the LPlm ARE modes and LPlm higher order core modes, respectively. Furthermore, an optical device (200) including the hollow-core fibre (100) and a method of manufacturing the hollow-core fibre are described.

The invention relates to a hollow-core fibre (HCF, or: hollow-corephotonic crystal fibre, HC-PCF) of non-bandgap type (or: hollow-coreanti-resonant-reflecting fibre, HC-AF), in particular having an axialhollow core region and an inner cladding region comprising anarrangement of anti-resonant elements (AREs) surrounding the coreregion. Furthermore, the invention relates to an optical deviceincluding at least one HC-AF and to a method of manufacturing a HCF ofnon-bandgap type. Applications of the invention are available in thefields of data transmission, in particular low-latency datatransmission, high-power beam delivery systems, in particular formaterial processing, modal filtering, gas-based non-linear optics, inparticular supercontinuum generation from the ultraviolet to theinfrared or generation of ultrashort pulses, fibre gyroscopes orchemical sensing.

In the present specification, reference is made to the following priorart illustrating the technical background of light-guiding fibres, inparticular HCF's of bandgap or non-bandgap type.

-   [1] T. A. Birks et al. in “Optics Letters” 1997. 22(13): p. 961-963;-   [2] N. A. Mortensen et al. in “Optics Letters” 2003. 28(20): p.    1879-1881;-   [3] US 2015/0104131 A1;-   [4] P. J. Roberts et al. in “Optics Express” 2005. 13(1): p.    236-244;-   [5] J. K. Lyngso et al. in “22nd International Conference on Optical    Fiber Sensors” Pts 1-3, 2012. 8421;-   [6] J. M. Fini et al. in “Nature Communications” 2014. 5: article    number 5085;-   [7] F. Couny et al. in “Optics Letters” 2006. 31(24): p. 3574-3576;-   [8] B. Debord et al. in “Optics Letters” 2014. 39(21): p.-   6245-6248;-   [9] W. Belardi et al. in “Optics Express” 2014. 22(8): p.    10091-10096;-   [10] W. Belardi et al. in “arXiv:1501.00586v2” 2015: p.    [physics.optics];-   [11] P. Jaworski et al. in “Optics Express” 2015. 23(7): p.    8498-8506;-   [12] A. Hartung et al. in “Optics Express” 2014. 22(16);-   [13] Benabid, A. F., et al., Hollow-core photonic crystal fibre, in    U.S. Pat. No. 8,306,379 B2, GLOphotonics SAS;-   [14] F. Poletti in “Optics Express” 2014. 22(20): p. 23807-23828;-   [15] T. G. Euser et al. in “Optics Express” 2008. 16(22): p.    17972-17981;-   [16] E. A. J. Marcatili et al. in “Bell Systems Technical    Journal” 1964. 43: p. 1783-1809;-   [17] J. M. Fini et al. in “Optics Express” 2013. 21(5): p.    6233-6242;-   [18] C. Wei et al. in “Optics Express” 2015. 23: p. 15824; and-   [19] A. Hartung et al. in “Optics Letters” 2015. 40(14): p. 3432.

Solid-core fibres are generally known and broadly, used e.g. in datacommunication applications. Solid-core fibres can be designed forlow-loss, single-mode transmission in a broadband transmission range ofthe fibre material, like quartz glass. So-called endlessly single-modeguidance (ESM, i.e. all higher order modes, HOMs, are leaky while thefundamental LP₀₁ mode is fully confined) is achieved in solid-corephotonic crystal fibres (PCFs) by engineering a cladding structuresurrounding the solid-core, as shown in FIG. 7A (prior art), such thatthe diameter d of channels in the cladding structure and theircentre-centre spacing (pitch) A fulfils the geometrical conditiond/Λ<0.41 ([1, 2]). However, due to light guiding in the solid fibrematerial, disadvantages exist in terms of increased latency in datatransmission, optically non-linear effects resulting in new lightfrequencies, and relatively low damage thresholds.

Through their ability to guide light in a non-solid core region, whichis evacuated (vacuum core), filled with a gas or filled with a liquid,hollow-core photonic crystal fibres, HC-PCFs, have unique advantagescompared to solid-core fibres, resulting in application areas such aslow-latency data transmission, high-power beam delivery, gas-basednon-linear optics, light guiding with ultralow nonlinearities andchemical sensing. HC-PCFs are typically divided into two classesdepending on the physical guidance mechanism: hollow-core photonicbandgap fibres (HC-PBFs) and hollow-core antiresonant-reflecting fibres(HC-AFs).

FIGS. 7B to 7I (prior art) shows a selection of scanning electronmicrographs of different types of conventional HC-PCFs. FIGS. 7B and 7Cshow HC-PBFs that confine modes inside a central hollow core by means ofa photonic bandgap in the cladding [4-6]. These types of PCF typicallyhave relatively low loss (ca. <20 dB/km) at telecommunicationwavelengths. However, due to the wavelength-specific effect of thephotonic bandgap, they guide light over a relatively narrow bandwidth(ca. <15 THz) only. Although in general HC-PBFs support HOMs, Fini etal. ([6], [3]) have shown that bended HC-PBFs can be made effectivelysingle-mode by including “satellite” hollow cores in the cladding (FIG.7B). These satellites strongly suppress HOMs in the core byphase-matching to them, causing high HOM loss. HC-PBGs can also be madetruly single-mode over a narrow spectral range (ca. <7 THz) if a smallenough core is used (see FIG. 7C) [5], but this results in fabricationdifficulties and significantly higher loss for the desired fundamentalmode.

FIGS. 7D to 7I show a selection of HC-AF structures, i. e. fibres havingguidance mechanism based mostly on anti-resonant effects. FIGS. 7D and7E have a Kagomé-lattice cladding [7, 8, 13] and FIGS. 7F and 7G haveone ring of single (F) or nested (G) anti-resonant elements (AREs). FIG.7H shows a HC-AF with a square core [12] and FIG. 7I depicts a HC-AFwith guiding properties in the ultraviolet [19]. Compared to HC-PBFs,the loss of HC-AFs is in general larger because of the non-idealconfinement, but the transmission window is broader.

F. Poletti showed numerically that by engineering the radial distancebetween nested AREs, a HC-AF can be made effectively single-mode over alimited wavelength range [14], but this has not been demonstratedexperimentally. In particular, mode suppression has been suggested in[14] for a certain centre wavelength and a certain structural parameterz/R only, wherein z is a diameter difference of AREs and nestedstructures in the AREs and R is the radius of the hollow core. Thistheoretical finding cannot be extended to a general design andmanufacturing of HC-AFs.

Also Wei et al. ([18]) and A. Hartung et al. ([19]) showed that HOMsuppression can be enhanced. This was achieved in HC-AFs with touchingAREs, but the overall HOM suppression was relatively low. According to[18], the effect of varying the thickness t of the touching AREs on theeffective refractive index of the hollow core and the AREs has beeninvestigated. An optimum thickness has been found wherein HOMs aresuppressed. As a disadvantage of this approach, HOM suppression has beenshown for a certain wavelength only, but not for a broad wavelengthrange.

Compared to solid-core fibres, one particular drawback of conventionalHC-AFs is their inability to be purely single-mode, i.e. higher ordermodes (HOMs) are supported for relatively long distances. As aconsequence the output beam quality is degraded, which is undesirable inmany applications as it introduces modal beating, blurring the focalspot and—if the fibre experiences change in stress—causes powerfluctuations. Another disadvantage of conventional HC-AFs results fromlimitation in manufacturing thereof, in particular positioning the AREsat specific azimuthal locations in a sufficient stable and reproduciblemanner.

The objective of the invention is to provide an improved hollow-corefibre of non-bandgap type, which is capable of avoiding disadvantages ofconventional HC-AFs. In particular, the HC-AF is to be provided with anextended wavelength range of single-mode transmission, increased loss ofHOMs, and/or increased ratio in loss between the highest-index core HOMand the fundamental, e. g. LP₀₁ mode. Furthermore, the objective of theinvention is to provide an improved optical device being equipped withat least one HC-AF and avoiding limitations of conventional opticaldevices, in particular in terms of low loss single mode guidance,damage-free high power light delivery and targeted creation of opticallynon-linear effects. Furthermore, the objective of the invention is toprovide an improved method of manufacturing a hollow-core fibre ofnon-bandgap type resulting in the improved inventive HC-AF and avoidinglimitations of conventional manufacturing methods.

These objectives are solved with a hollow-core fibre of non-bandgaptype, an optical device and a method of manufacturing a hollow-corefibre of non-bandgap type, comprising the features of the independentclaims, resp. Advantageous embodiments and applications of the inventionare defined in the dependent claims.

According to a first general aspect of the invention, the aboveobjective is solved by a hollow-core fibre of non-bandgap type,comprising a hollow core region axially arranged along the longitudinalextension of the hollow-core fibre and an inner cladding regioncomprising an arrangement of multiple anti-resonant elements, AREs,surrounding the core region along the length of the hollow-core fibre.The core region is adapted for guiding a transverse fundamental coremode and transverse higher order core modes of a light field coupledinto the HC-AF. The core region has a smallest transverse core dimension(D), which is the smallest distance between AREs on diametricallyopposite sides of the core region. Preferably, the smallest transversecore dimension is constant along the length of the HC-AF. Each of theAREs is adapted for guiding transverse ARE modes, and the i-th ARE has asmallest transverse ARE dimension (d_(i)). Preferably, the smallesttransverse ARE dimension also is constant along the length of the HC-AF.The core region and the AREs are configured to provide phase matching ofthe higher-order core modes of the core region to the ARE modes of theAREs. In other words, the higher order core modes and the ARE modes haverefractive indices which are equal or approximated to each other so thatthe higher-order core modes can resonantly couple to the ARE modes, i.e.the ARE modes can be excited by the higher-order core modes.

The hollow core region of the inventive HC-AF is formed by an innerspace of the hollow-core fibre, which is empty (evacuated) or filledwith a gas, in particular at least one of air, noble gas and hydrogen, aliquid, and/or a material having a non-linear optical response, likee.g. at least one of the above gases or a Rydberg gas. Accordingly, theterm “hollow core region” covers any longitudinal waveguide structurewhich is free of solid materials in its inner space. The core region hasa first refractive index determined by the material thereof, while therefractive (or effective) index of the core mode is determined by thematerial refractive index, the shape of the core region and theexcitation wavelength.

In radial directions, the core region is surrounded by the AREs, whichare arranged in a non-touching manner. In other words, the innercladding region comprises an arrangement of single AREs without anazimuthal contact thereof. The term “anti-resonant element”, ARE (or:“tube” or “cladding capillary”), refers to any hollow wave-guidingelement having a smaller diameter than the hollow core region andextending along the length of the HC-AF. Preferably, the wall thicknessof the AREs of the inner cladding region is smaller than 20% of the coredimension D, e. g. <5 μm. All AREs can have the same smallest innerdimension, or each ARE can have a different smallest inner dimension.The inner cladding region surrounding the core region has a refractiveindex (a second refractive index) higher than the first refractive indexof the core region. The hollow AREs are empty (evacuated) or filled likegas or liquid, e. g. like the core region.

An even number of AREs, e. g. 4 or 6, or an odd number of AREs, e. g. 3or 5 or 7, is provided, which can be arranged with an even orodd-numbered symmetry delimiting the core region. Preferably, the AREsare non-nested AREs, and/or the AREs are made of a glass, in particularsilica, a plastic material, in particular polymer, composite, inparticular ZBLAN fiber composite, metal or crystalline material.Preferably, for holding the AREs, an outer cladding region is provided,wherein the AREs are attached to an inner surface of the outer claddingregion.

According to the invention, each of the ARE dimensions (d_(i)) and thecore dimension (D) are selected such that a ratio of the ARE and coredimensions (d_(i)/D) is approximated to a quotient of zeros of Besselfunctions of first kind (u_(lm,ARE)/u_(lm,core)), multiplied with afitting factor in a range of 0.9 to 1.5, preferably 0.96 to 1.20, with mbeing the m-th zero of the Bessel function of first kind of order l,said zeros of the Bessel functions describing (modelling) the LP_(lm)ARE modes and LP_(lm) higher order core modes, respectively.Advantageously, the inventors have found a modal filtering effectdepending on only one dimensionless geometrical parameter (d_(i)/D),akin to the well-known d/A parameter for endlessly single-modesolid-core PCF.

The inventors have found that modal refractive indices of the LP_(lm)ARE modes and LP_(lm) higher order core modes can be represented by ananalytical model based on modes of coupled capillary waveguides, saidmodes being approximated by a Marcatili-Schmeltzer expression, whichdepends on the zeros of the Bessel function providing the modes and thefitting factor. Phase-matching the LP_(lm) ARE modes and LP_(lm) higherorder core modes, i. e. matching the modal indices thereof can beobtained if the ratio of the ARE and core dimensions (d_(i)/D) isadapted to the quotient of zeros of Bessel functions of first kind(u_(lm,ARE)/u_(lm,core)), multiplied with the fitting factor.

Preferably, the fitting factor is a model parameter, which is obtainedfrom matching the analytical model to a vectorial finite element (FE)modelling of the LP_(lm) ARE modes and LP_(lm) higher order core modesof the HC-AF. In particular, the fitting factor results from a divisionfrom a first factor matching the analytical model to the FE model of theLP_(lm) higher order core modes and a second factor matching theanalytical model to the FE model of the LP_(lm) ARE modes, as furtherdescribed below with reference to equation (4).

The terms “approximating” (or “adapting”) include an equality of theratio of the ARE and core dimensions and the quotient of zeros of Besselfunctions of first kind, multiplied with the fitting factor, or adifference between the ratio of the ARE and core dimensions and thequotient of zeros of Bessel functions of first kind, multiplied with thefitting factor, being minimized such that a ratio in loss (dB/m) betweenthe highest-index core HOM and the LP₀₁ core mode is >5, inparticular >10 or even >25, preferably >50.

Advantageously, with the above condition for selecting the ratio of theARE and core dimensions, the inventors have found a new design parameterfor the HC-AFs allowing a broadband extension of the HOM coupling to AREmodes, e.g. up to all wavelengths within the transparency window of itsfundamental core mode. According to the invention, refractive indices ofthe core region and the AREs are equal or approximated to each other ina broad wavelength range, preferably covering at least 10 THz,particularly preferred covering at least 20 THz.

All modes except the fundamental are strongly suppressed, resulting ineffectively single-mode behaviour of the HC-AF (effectively endlesslysingle-mode behaviour, eESM). Contrary to (effectively) single-modeHC-PBFs [17], the proposed HC-AF structure provides a much largerbandwidth at relatively low loss while providing eESM behaviour.Furthermore, contrary to the HC-AF described in [18], wherein thethickness of the glass capillaries in the cladding has been optimizedonly, the invention provides the new design parameter for the ratio ofthe ARE and core dimensions. This new design provides realphase-matching, whereas in [18] phase-matching was not achieved (seeFIG. 2a of [18]). Contrary to the SM guidance for HC-PBFs [6] whereshunts have been included in the bandgap cladding to enhance HOMsuppression for a relatively narrow wavelength range, the inventiveHC-AF supports HOM suppression for an extended frequency range.

The ratio of the ARE and core dimensions is selected on the basis of amodel which has been found by the inventors for describing tubular AREs(ARE capillaries) with a circular transverse cross-section as outlinedbelow. In this case, the smallest transverse ARE dimension (d_(i)) isthe inner diameter of the ARE. As a further advantage of the invention,the selected design parameter is valid not only for tubular AREs, butalso for AREs having other cross-sections, like elliptical or evenpolygonal cross-sections. With the latter variants, the smallesttransverse ARE dimension (d_(i)) is the smallest inner cross-sectionaldimension of the ARE. The inventors have found that the model based ontubular AREs still provides an excellent approximation for HOMsuppression in non-circular AREs, if they can be approximately modelledlike a tubular ARE.

According to a preferred embodiment of the invention, the AREs have afirst smallest transverse ARE dimension (d_(i)) and the ratio of thefirst smallest transverse ARE dimension and the core dimension (d_(i)/D)is approximated to a quotient of zeros of Bessel functions of first kind(u_(01,ARE)/u_(11,core)), multiplied with the fitting factor, said zeros(u_(01,ARE)), (u_(11,core)) describing the LP₀₁ ARE modes and the LP₁₁core mode, respectively. Accordingly, the most pronounced HOM of thecore region, the LP₁₁ core mode, is coupled to the ARE modes. With thisembodiment, preferably all AREs have the same smallest transverse AREdimension. As an example, all AREs are tubular AREs having the sameinner diameter. According to particularly preferred variants of thisembodiment, the ratio of the first ARE dimension and the core dimension(d₁/D) is selected in a range from 0.5 to 0.8, preferably from 0.60 to0.75, in particular in a range from 0.62 to 0.74. Advantageously, aHC-AF is provided with these ranges, having low-loss guidance for thefundamental mode at all wavelengths within its transmission window,strongly suppressing the LP₁₁ core mode and also all higher-order modes.

According to a further advantageous embodiment of the invention, theAREs comprise a first group of AREs with the first ARE dimension (d₁)and a second group of AREs with a second smallest transverse AREdimension (d₂) smaller than the first ARE dimension (d₁) of the firstgroup of AREs. With this embodiment, the ratio of the second AREdimension and the core dimension (d₂/D) is approximated to a quotient ofzeros of Bessel functions of first kind (u_(01,ARE)/u_(21,core)),multiplied with the fitting factor, said Bessel functions (u_(01,ARE)),(u_(21,core)) describing the LP₀₁ ARE modes and the LP₂₁ core mode,respectively. AREs having the first ARE dimension and AREs having thesecond ARE dimension can be arranged in an alternating fashion,surrounding the hollow core region of the HC-AF.

According to particularly preferred variants of this embodiment, theratio of the second ARE dimension and the core dimension (d₂/D) isselected in a range from 0.3 to 0.7, preferably from 0.43 to 0.59, inparticular in a range from 0.45 to 0.54. Advantageously, a HC-AF isprovided with these ranges, having a further improved suppression of theHOMs.

Advantageously, multiple variants are available for designing thearrangement of AREs, which provide the inner cladding region of theinventive HC-AF. These variants can be selected as alternatives or incombination in dependency on the particular application of theinvention. Firstly, the arrangement of AREs can have a three-foldsymmetry. Alternatively, the arrangement of AREs can have two-foldsymmetry and causes optical birefringence. Furthermore, the AREs can bearranged such that the cross-sections thereof are distributed on asingle ring surrounding the core region. Alternatively, the AREs can bearranged such that the cross-sections thereof are distributed onmultiple, e. g. two, three or more, coaxial rings surrounding the coreregion. Advantageously, lower loss HC-AFs with improved HOMS suppressioncan be realized with these variants of the invention.

The outer cladding region for holding the AREs has the shape of a hollowsleeve extending along the HC-AF and having an inner surface forattaching the AREs and an outer surface which can be exposed as thesurface of the HC-AF or covered with further layers, e. g. protectivelayers, like a polymer coating, or an opaque layer. According to avariant of the invention, the outer cladding region has an innertransverse cross-section with a regular polygonal shape, e.g. ahexagonal shape, and the AREs are fixed to corners of the polygonalshape. As an advantage, each ARE is fixed along two contact lines withthe inner surface of the outer cladding region, thus increasing themechanical stability of the inner cladding region (AREs) when preparingthe preform. Furthermore, the AREs are fixed with a regular azimuthalspacing by arranging them in the corners of the polygonal shape.According to an alternative variant of the invention, the outer claddingregion has an inner transverse cross-section with a curved, inparticular circular shape, and the AREs are evenly distributed in thecurved shape.

According to a second general aspect of the invention, the aboveobjective is solved by an optical device, including at least onehollow-core fibre according to the above first general aspect of theinvention. Preferably, the optical device comprises at least one of amodal filtering device, a light source, in particular a laser, anoptical amplifier, a beam delivery system, a data communication system,a frequency converter, in particular for supercontinuum generation and apulse shaper, in particular for pulse compression.

According to a third general aspect of the invention, the aboveobjective is solved by a method of manufacturing a hollow-core fibreaccording to the above first general aspect of the invention.Preferably, the method of manufacturing a hollow-core fibre comprisesthe steps of providing ARE preforms and a hollow jacket preform, fixingthe ARE preforms on an inner surface of the jacket preform in adistributed manner, and heating and drawing the jacket preform includingthe ARE preforms until the final ARE and core dimensions are set.

Optionally, the jacket preform including the ARE preforms are—in a firstheating and drawing step—first drawn to a cane and—in a second heatingand drawing step—then drawn to a fibre until the final ARE and coredimensions are set.

Preferably, the heating and drawing steps include applying a vacuum oran increased fluid pressure to at least one of the jacket preform andthe ARE preforms for setting the ARE and core dimensions, respectively.Advantageously, this allows a precise adjustment of the smallesttransverse core dimension (D) of the core region and the smallesttransverse ARE dimension(s) (d_(i)) of the AREs.

According to a preferred embodiment of the invention, the ARE and coredimensions are selected, preferably set during the final heating anddrawing step, such that the phase matching of the higher order coremodes and the ARE modes is obtained over a frequency range above 20 THz.This can be achieved by theoretical considerations or referencemeasurements.

According to further advantageous embodiments of the invention, apost-processing step can be provided, wherein at least one of the coreregion and the anti-resonant elements can be filled with at least one ofa gas, in particular air, noble gas and/or hydrogen, a liquid, and/or amaterial having a non-linear optical response. To this end, at least oneportion of the hollow core fiber can be enclosed into a cell. The celland all of the hollow core regions can be filled with a material from anexternal reservoir, or some of the hollow core regions can be filledwith a material from a first external reservoir and some of anothermaterial of a second external reservoir. This post-processing step canbe implemented just before or during the application of the fiber, e.,g. in an optical experiment. Further post-processing steps can followfor adapting the inventive fibre to the application thereof, e. g. indata transmission or non-linear optics.

Further advantages and details of the invention are described in thefollowing reference to the attached drawings, which show in:

FIG. 1: cross-sections of HC-AFs according to preferred embodiments ofthe invention;

FIG. 2: graphical illustrations of coupling HOMs of the core region withARE modes and numerical simulations when varying d/D to find the optimumratio between ARE dimension d and core dimension D;

FIG. 3: graphical illustrations of numerical simulations to illustratethe scalability of the inventive HC-AF;

FIG. 4: illustrations of applying the inventive model for designing aHC-AF;

FIG. 5: an optical device according to an exemplary embodiment of theinvention;

FIG. 6: steps of a method of manufacturing an inventive HC-AF; and

FIG. 7: cross-sections of conventional solid or hollow core fibres(prior art).

In the following, exemplary reference is made to choosing a propergeometry of the HC-AF, in particular the diameter of the core and AREs,like glass capillaries in the inner cladding region. Implementing theinvention is not restricted to the indicated examples of geometricquantities, like the dimensions D, d and t, but rather possible withvaried values providing the inventive design parameters. Features ofHC-AFs, the manufacturing thereof and the application thereof are notdescribed, as far as they are known from prior art.

Practical Examples of HC-AFs

FIGS. 1A to 1E show transverse cross-sections of practical examples ofinventive HC-AFs 100 (perpendicular to the axial. extension thereof).The bright circles represent the solid material of the AREs or outercladding region, like quartz glass or silica, while the dark portionsare free of solid materials (evacuated or filled with gas or liquid).The geometric design of the HC-AFs 100 is selected as outlined in themodel section below.

Each HC-AF 100 comprises a hollow core region 10 (represented in FIG. 1Aby a dotted circle), an inner cladding region 20 with multiple AREs 21and an outer cladding region 30. The hollow core region 10 is the emptyspace, between the AREs 21, extending along the longitudinal length ofthe HC-AF 100 and having a smallest transverse core dimension D. TheAREs 21 of the inner cladding region 20 comprise capillaries having awall thickness t and a smallest transverse ARE dimension d. The AREs 21are fixed to the inner surface of the outer cladding region 30, e.g. asdescribed below with reference to FIG. 6. The outer cladding region 30comprises a larger capillary being made of e.g. glass and providing aclosed cladding of the HC-AF 100.

HC-AF 100 of FIG. 1A illustrates an embodiment wherein the AREs 21comprise a single-ring of six thin-wall capillaries with a circulartransverse cross-section (inner diameter d=13.6 μm and wall thicknesst=0.2 μm) arranged within the larger capillary of the outer claddingregion 30 in six-fold symmetric pattern so as to create a central hollowcore of diameter D (the shortest distance between diametrically oppositeAREs 21), with D=20 μm. The outer cladding region 30 has an outerdiameter of 125 μm and a cladding thickness of 38 μm. Alternatively, thecore dimension D can be selected in a range from 10 μm to 1000 μm,wherein the other geometrical parameters (like d, t) are scaledaccordingly.

FIG. 1B shows a modified embodiment with multiple, in particular twocoaxial rings of AREs 21 (d=13.6 μm, t=0.2 μm, and D=20 μm arrangedwithin the outer cladding region 30 with six-fold symmetry. For holdingthe inner and outer rings of AREs 21, a support tube 22 is included inthe HC-AF 100. The support tube 22 is made of e.g. silica with adiameter of e.g. 48 μm.

FIGS. 1C and 1D illustrate a three-fold symmetry and FIG. 1E shows atwo-fold symmetry of the inventive HC-AF 100, advantageously offeringenhanced eESM effects.

According to FIG. 1C, the AREs 21A, 21B comprise a first group of AREs21A with a first, larger ARE dimension d_(i) (e. g. 13.6 μm) and asecond group of AREs 21B with a second, smaller ARE dimension d₂ (e. g.10.2 μm), both with a wall thickness of e.g. 0.2 μm. On the innersurface of the outer cladding 30, longitudinal protrusions 31 areprovided, which have an axial extension along the HC-AF 100. In thecross-sectional view of FIG. 1C, the protrusions 31 are shown as blobs.Preferably, the protrusions 31 are made of the same material like theAREs, e. g. glass. Each of the smaller AREs 21B is fixed to the top ofone of the protrusions 31. The AREs 21A, 21B surround the core region 10having a diameter D=20 μm. The radial height of the protrusions 31preferably is selected so as to yield the most circularly-symmetricguided mode in the central core region 10. Alternatively, theprotrusions 31 could be omitted, yielding a simpler structure, as shownin FIG. 1D.

By arranging the AREs 21A, 21B so as to form a two-fold symmetricstructure (FIG. 1E), a birefringent polarization-maintaining HC-AF isobtained, displaying eESM behaviour.

The examples of inventive HC-AFs 100 as shown in FIG. 1 can be modified,in particular with regard to the shape of the AREs 21, 21A, 21B, whichcan have e. g. an elliptic or polygonal cross-section; the inner shapeof the outer cladding 30, which can have e. g. a polygonal cross-section(see FIG. 6); the solid materials of the AREs 21, 21A, 21B, which maycomprise e. g. plastic material, like PMMA, glass, like silica, orsoft-glass, like ZBLAN; the dimensions of the AREs 21, 21A, 21B; thenumber of rings of AREs, e. g. three or more; the number of AREs, e. g.4 or 5 or 7 or more; and the symmetry of the ARE arrangement.

Model for HC-AF Design

The ARE dimension (d_(i)) and the core dimension (D) of the inventiveHC-AFs 100 are selected such that a ratio of the ARE and core dimensions(d_(i)/D) is approximated to a quotient of zeros of Bessel functions offirst kind (u_(lm,ARE)/u_(lm,core)), multiplied with the fitting factor,as defined above. If all AREs have the same ARE dimension (d₁), theratio of the ARE dimension and the core dimension (d₁/D) preferably isapproximated to a quotient of zeros of Bessel functions of first kind(u_(01,ARE)/u_(11core)), multiplied with the fitting factor, wherein thezeros (u_(01,ARE)), (u_(11,core)) describe the LP₀₁ ARE modes and theLP₁₁ core mode, respectively. If further AREs have a second, smaller AREdimension, the ratio of the second ARE dimension and the core dimension(d₂/D) preferably is approximated to a quotient of zeros of Besselfunctions of first kind (u_(01,ARE)/u_(21,core)), multiplied with thefitting factor, wherein the Bessel functions (u_(01,ARE)), (u_(21,core))describe the LP₀₁ ARE modes and the LP₂₁ core mode, respectively.

These design conditions are found on the basis of the theoreticalconsiderations and numerical simulations illustrated in the followingwith reference to FIGS. 2 to 4. These theoretical considerations andnumerical simulations can be correspondingly extended to the coupling ofcore modes higher than the LP₂₁ core mode to the ARE modes.

The central core region 10 of the HC-AF 100 supports several transversecore modes each with a characteristic modal refractive index and leakageloss. The inventive structure is provided in such a way that the LP₀₁mode (with the highest effective index) has a loss that is much lowerthan any of the core HOMs. This is achieved by designing the AREs 21,21A, 21B and the gaps between them so that they support a band of leakymodes (or states) that phase-match to HOMs in the core region 10, makingthem highly leaky. This strong loss discrimination can be made broadbandenough for obtaining eESM behaviour.

FIG. 2A shows the HC-AF 100 of FIG. 1A (left) illustrating thefundamental LP₀₁ core mode (centre) and the LP₁₁ core mode (right) aswell as the leaky ARE modes (right), to which the LP₁₁ core mode andhigher core modes are resonantly coupled. FIG. 2B shows curves of thefinite-element (FE) modelling computed effective refractive index independency on the parameter d/D, and FIG. 2C shows curves of leakageloss for changing ARE dimension d but constant core dimension D. Thecentral curve in FIG. 2C shows the corresponding HOM suppression. Thedashed lines show the fully vectorial computed values for the LP₀₁ modeof a free standing tube with diameter d and thickness t. Exemplarystructure parameters are t/D=0.01 and D/λ=20, n_(glass)=1.45 andn_(core)=1.

The structure shown in FIG. 2A supports several leaky transverse coremodes formed by anti-resonant reflection at the walls of the AREs 21with negative curvature. The AREs 21 support a cladding photonicbandstructure provided by the inner cladding region 20, and they can beapproximated by quasi free-standing, thin-walled. tubes. The thicksurrounding glass wall of the outer cladding region 30 does notappreciably influence the modal properties, but is provided tophysically support the AREs 21.

FIG. 2B shows the effective index distribution of the two highest-indexcore modes LP₀₁ and LP₁₁ when varying the ARE diameter (for fixed D),effectively changing the cladding photonic bandstructure. The index ofthe LP₀₁ core mode is high enough to avoid resonant coupling to thefundamental ARE modes, and remains almost independent of d/D, whereasthe LP₁₁ core mode undergoes a strong anti-crossing with the fundamentalARE mode at d/D≈0.68. As one moves away from this anti-crossing, theeven and odd eigenmodes evolve asymptotically into uncoupled LP₁₁ coreand fundamental ARE modes. Core modes of even higher order (not shown inFIG. 2B) have lower indices and couple to highly leaky modes of the AREring, some of which are concentrated in the gaps between the AREs 21.

FIG. 2C plots the calculated leakage loss of the LP₀₁ mode and the twohybrid LP₁₁/ARE₀₁ modes. Over the range shown the LP₀₁ core mode has arelatively constant loss with a minimum value of 0.17 dB/m at d/D 0.65.For smaller ARE 21 diameters the loss increases, closely matching thevalue for an isolated thick-walled dielectric capillary in the limitd/D→0 (not shown in FIG. 2C). This limit was used to cross-check the FEcalculations with analytical results [16], in particular the accuracy ofthe perfectly matched layers (PMLs). At the anti-crossing point the lossof the two hybrid LP₁₁/ARE₀₁ modes strongly increases, almost reachingthe value for an isolated capillary in vacuum (dashed brown line), whichwas calculated by solving Maxwell's equations in full vectorial form.This provides further confirmation that the PMLs were set up correctly.

The HOM suppression increases strongly at the anti-crossing, peaking ata value of about 1200. Far from the anti-crossing it drops to less than5, which is similar to values typically achieved in kagomé-PCF [15]. Fora comprehensive analysis, the HOM suppression of all the higher-ordercore modes must be calculated. FE modelling reveals that the HOM withthe next-lowest loss after the LP₁₁ core mode is the four-lobed LP₂₁core mode, with a HOM suppression of ˜70 at d/D≈0.68 and ananti-crossing with the fundamental ARE mode at d/D≈0.51. In experiments,however, this particular core mode is less likely to be excited byend-fire illumination or by stress- and bend-induced scattering from theLP₀₁ core mode (the index difference is some two times larger than forthe LP₁₁ core mode). FE modeling shows that LP_(lm) core modes of evenhigher order do not affect the overall HOM suppression because theyphase-match to modes of the ARE ring (some of which are concentrated inthe gaps between the AREs 21), resulting in strong leakage loss.

FIG. 3A plots, versus D/λ (λ: wavelength), the difference Δn_(lm)between the refractive indices of the LP_(lm) core modes and thefundamental ARE mode at constant d/D=0.68 and t/D=0.01. Δn₀₁ decreaseswith increasing D/λ but overall remains positive. As a consequence, theLP₀₁ core mode is anti-resonant with the fundamental ARE mode andremains confined to the core (see also left panel of the inset in FIG.3A). In contrast, Δn₁₁ is much smaller, reaching values as small as 10⁻⁶at D/λ≈66.

At certain values of D/λ, anti-crossings appear between the LP₀₁ modeand the q-th order transverse mode in the glass walls of the AREs 21,following the simple relationship:

$\begin{matrix}{( \frac{D}{\lambda} )_{q} \approx \frac{q}{2( {t\text{/}D} )\sqrt{n_{g}^{2} - 1}}} & (1)\end{matrix}$

The vertical dotted lines in FIG. 3 are centred at the first two ofthese resonances, for t/D=0.01. In the vicinity of these points the LP₀₁core mode leaks rapidly through the resonant AREs 21 into the solidglass jacket 30, yielding loss values that are close to those of anisolated thick-walled dielectric capillary [16]; the result is a strongreduction in the HOM suppression (see FIG. 3B). Away from these narrowregions, however, the HOM suppression remains relatively high—aconsequence of the fact that the indices of the LP₁₁ core andfundamental ARE modes remain close to each other. The result is verystrong LP₁₁ core mode suppression over all the ranges of LP₀₁ modetransmission.

To explain why maximum HOM suppression occurs at d/D=0.68 for allwavelengths (except in the vicinity of ARE wall resonances, see Eq. 1),the inventors have applied an analytical model in which the realstructure with the core 10 and the AREs 21 are treated as thick-walledcapillaries (see FIG. 4A). The modal indices of the LP_(1m) modes in athick-walled capillary can be approximated by the modifiedMarcatili-Schmeltzer expression [16]:

$\begin{matrix}{n_{lm} = \sqrt{1 - {( \frac{u_{lm}}{\pi \; f_{s}} )^{2}( \frac{\lambda}{d_{l}} )^{2}}}} & (2)\end{matrix}$

where u_(lm) is the m-th zero of the Bessel function J_(l) and d_(i) isthe inner diameter of the capillary. The parameter f_(s) (which has avalue close to 1, s=co represents the core 10 and s=ARE the AREs 21) isused to heuristically fit the analytical values from the model equationto the results of FE simulations. It corrects for the non-circular coreand the finite wall thicknesses of core 10 and AREs 21.

FIG. 4B plots the effective index of the two LP₁₁/ARE₀₁ hybrid modestogether with the fitted values for the LP₁₁ mode (zero line) computedusing Eq. (2) with fit parameters f_(co)=1.077 for the core 10 andf_(ARE)=0.990 for the ARE 21. The convenient analytical form of Eq. (2)allows one to derive a simple expression for the d/D value at which theLP₁₁ core and ARE₀₁ modes couple optimally:

$\begin{matrix}{\frac{d}{D} = {{\frac{u_{01}}{u_{11}}\frac{f_{co}}{f_{ARE}}} = 0.68}} & (3)\end{matrix}$

Eq. (3) provides a convenient rule-of-thumb for designing robustlysingle-mode eESM PCFs. To a first approximation it depends neither onthe refractive indices nor on the absolute physical dimensions of thefibre, making the design scalable. This means that, provided the ratiod/D is maintained, it becomes possible to design large-core eESM PCFsand to deliver losses of some dB/km in multiple transmission windows,the broadest of which spans more than one octave.

By using Eq. (2) one can also easily find structural parameters wherehigher order core modes (e.g. the LP₂₁ core mode) are effectivelysuppressed. Also by adjusting the physical dimensions, the resonancebands can be blue/red shifted (for smaller/thicker wall thickness t) andthe minimum transmission loss of the LP₀₁ core mode can be adjusted (forchanging core diameter).

Eq. (2) can be also used to find appropriate geometrical parameters fordesigning a HC-AF with an enhanced eESM effect, i.e., a fibre where thefirst two HOMs of the core couple to resonances in the AREs. This yieldsthe conditions:

$\begin{matrix}{\frac{d_{1}}{D} = {{\frac{u_{01}}{u_{11}}\frac{f_{co}}{f_{ARE}}} = {{0.68\mspace{14mu} {and}\mspace{14mu} \frac{d_{2}}{D}} = {{\frac{u_{01}}{u_{21}}\frac{f_{co}}{f_{ARE}}} = 0.51}}}} & (4)\end{matrix}$

A fibre structure with such an enhanced eESM effect is depicted in FIGS.1C to 1E consisting of one or several of single rings of AREs withdifferent inner diameters d₁ and d₂.

Embodiments of Optical Device

The inventive HC-AF 100 has multiple applications for light guiding,e.g. for beam delivery, data transmission or frequency conversionpurposes. Accordingly, an optical device, which represents a furthersubject of the invention, comprises at least one inventive HC-AF 100 andfurther optical components, monitoring components, detector componentsand/or control components, which are selected in dependency on theparticular application of the optical device.

FIG. 5 schematically illustrates an optical device 200 according to anembodiment of the invention, which is adapted for a high power beamdelivery, e.g. for surface processing purposes. The optical device 200comprises a light source 210, like a laser source, and the HC-AF 100.The output of the light source 210 is optically coupled with an inputside of the HC-AF 100, while an output side thereof is directed to alocation of beam delivery (see arrow).

With alternative applications of the invention, the light source 210comprises a laser source for driving a frequency conversion process, inparticular a supercontinuum generation process or pulse compression,inside the HC-AF 100. According to yet further applications, the lightsource 210 may comprise an optical transmitter of a data communicationsystem, which is coupled via the HC-AF 100 with an optical receiver (notshown).

It is noted that FIG. 5 represents a schematic drawing only. Details ofan optical device including at least one inventive HC-AF 100 can beimplemented as it is known from conventional optical devices.

Method of Manufacturing HC-AFs

FIG. 6 schematically illustrates the main steps of manufacturing aninventive HC-AF 100. FIG. 6 is a schematic illustration only, whichpresents the main steps of providing ARE preforms 23 and a hollow jacketpreform 32 (FIG. 6A), fixing the ARE preforms 23 on an inner surface ofthe jacket preform 32 (FIG. 6B) and heating and drawing the preformjacket 32 including the ARE preforms 23 for obtaining the HC-AF 100(FIG. 6C). Optionally, the heating and drawing step may include a firststep of heating and drawing the jacket preform 32 with the ARE preforms23 to a cane, and a second step of heating and drawing the cane untilthe ARE and core dimensions are set. Details of the steps in FIGS. 6A,6B and 6C can be implemented as far as they are known from conventionalfibre manufacturing methods.

According to FIG. 6A, the jacket preform 32 is a hollow tubular form,made of glass, which has an inner transverse cross-sectional regularhexagonal shape. The outer diameter of the preform jacket 32 is e.g. 28mm, while the inner transverse dimension is about 22 mm. Thelongitudinal length of the preform jackets 32 and the ARE preforms 23 isabout 120 cm.

With the fixing step of FIG. 6B, the ARE preforms 23 are fixed to thecorners of the inner hexagonal shape of the jacket preform 32. This isobtained by applying heat resulting in a physical connection between AREpreforms and jacket preform. Subsequently, the composite of the jacketpreform 32 and the ARE preforms 23 is drawn during the application ofheat until the ARE and core transverse dimensions are obtained. The AREand core transverse dimensions can be influenced by applying a vacuum oran increased pressure to the jacket preform 32 and/or the ARE preforms23 during the heating and drawing steps.

Applying a vacuum or an increased pressure during the heating anddrawing steps is schematically illustrated in FIG. 6B. The hollow innerspace of the jacket preform 32 is connected with a first externalreservoir 41, like a source of pressurized nitrogen. Furthermore, theARE preforms 23 are connected with at least one second externalreservoir 42, like e.g. an external source of pressurized nitrogen. Ifall AREs are to be produced with the same inner transverse dimension,all AREs can be connected with a common external reservoir. Otherwise,e.g. two groups of AREs are connected to two different externalreservoirs for creating different inner transverse dimensions, as showne.g. in FIG. 1E. The final heating and drawing step is conducted e.g. ina furnace selected in dependency of the material of the jacket preform32 and the ARE preforms 23 (e.g. around 2000° C. for silica components).

After obtaining the final HC-AF 100, it can be filled with a gas, likeair or a noble gas or hydrogen, or a liquid, like water, and the inputand output sides of the HC-AF 100 are enclosed by a cell withstandinghigh fluidic pressure and which is partially transmissive, e.g. byincluding a glass plate, for optical radiation, e.g. from a lasersource.

The features of the invention disclosed in the above description, thedrawings and the claims can be of significance individually, incombination or sub-combination for the implementation of the inventionin its different embodiments.

1-27. (canceled)
 28. A hollow-core fibre of non-bandgap type,comprising: a hollow core region axially extending along the hollow-corefibre and having a smallest transverse core dimension (D), wherein thecore region is configured for guiding a transverse fundamental core modeand a plurality of transverse higher order core modes, and an innercladding region including an arrangement of anti-resonant elements(AREs) surrounding the core region along the hollow-core fibre, eachhaving a smallest transverse ARE dimension (d_(i)) and being configuredfor guiding a plurality of transverse ARE modes, wherein the core regionand the AREs are configured to provide phase matching of the higherorder core modes and the ARE modes, and the ARE dimension (d_(i)) andthe core dimension (D) are selected such that a ratio of the ARE andcore dimensions (d_(i)/D) is approximated to a quotient of zeros ofBessel functions of first kind (u_(lm,ARE)/u_(lm,core)), multiplied witha fitting factor in a range from 0.9 to 1.5, with m being the m-th zeroof the Bessel functions of first kind of order l, said zeros of theBessel functions describing the LP_(lm) ARE modes and LP_(lm) higherorder core modes, respectively.
 29. The hollow-core fibre according toclaim 28, wherein the AREs surrounding the core region are arranged in anon-touching manner.
 30. The hollow-core fibre according to claim 28,wherein the AREs have a first smallest transverse ARE dimension (d₁),and the ratio of the first ARE dimension and the core dimension (d₁/D)is approximated to a quotient of zeros of Bessel functions of first kind(u_(01,ARE)/u_(11core)), multiplied with the fitting factor, said zeros(u_(01,ARE)), (u_(11,core)) describing the LP₀₁ ARE modes and the LP₁₁core mode, respectively.
 31. The hollow-core fibre according to claim30, wherein the ratio of the first ARE dimension and the core dimension(d₁/D) is selected in a range from 0.5 to 0.8.
 32. The hollow-core fibreaccording to claim 31, wherein the ratio of the first ARE dimension andthe core dimension (d₁/D) is selected in a range from 0.62 to 0.74. 33.The hollow-core fibre according to claim 30, wherein each of the AREshas the first ARE dimension (d₁).
 34. The hollow-core fibre according toclaim 30, wherein a first group of the AREs has the first ARE dimension(d₁), and a second group of the AREs has a second smallest transverseARE dimension (d₂) smaller than the first ARE dimension (d₁) of thefirst group of the AREs, and the ratio of the second ARE dimension andthe core dimension (d₂/D) is approximated to a quotient of zeros ofBessel functions of first kind (u_(01,ARE)/u_(21,core)), multiplied withthe fitting factor, said Bessel functions (u_(01,ARE)), (u_(21,core))describing the LP₀₁ ARE modes and the LP₂₁ core mode, respectively. 35.The hollow-core fibre according to claim 34, wherein the ratio of thesecond ARE dimension and the core dimension (d₂/D) is selected in arange from 0.3 to 0.7.
 36. The hollow-core fibre according to claim 34,wherein the ratio of the second ARE dimension and the core dimension(d₂/D) is selected in a range from 0.45 to 0.54.
 37. The hollow-corefibre according to claim 28, wherein the number of AREs is 3, 4, 5, 6 or7.
 38. The hollow-core fibre according to claim 28, wherein thearrangement of AREs has at least one of the features: the arrangement ofAREs has three-fold symmetry, the arrangement of AREs has two-foldsymmetry and causes optical birefringence, the AREs are arranged suchthat the cross-sections thereof are distributed on a single ringsurrounding the core region, and the AREs are arranged such that thecross-sections thereof are distributed on multiple rings surrounding thecore region.
 39. The hollow-core fibre according to claim 28, whereinthe AREs have at least one of the features: the AREs have a circular,elliptic or polygonal transverse cross-section, and the AREs are made ofone of a glass, polymer, composite, metal and crystalline material. 40.The hollow-core fibre according to claim 28, wherein at least one of thecore region and the AREs is evacuated or filled with at least one of agas, a liquid, and a material having a non-linear optical response. 41.The hollow-core fibre according to claim 28, further comprising an outercladding region surrounding the inner cladding region along thehollow-core fibre, wherein the outer cladding region has a transversecross-section with a polygonal shape, and the AREs are located incorners of the polygonal shape.
 42. The hollow-core fibre according toclaim 28, further comprising an outer cladding region surrounding theinner cladding region along the hollow-core fibre, wherein the outercladding region has a transverse cross-section with a curved, and theAREs are evenly distributed in the curved shape.
 43. The hollow-corefibre according to claim 28, wherein the core region and the AREs areconfigured to provide the phase matching of the higher order core modesand the ARE modes in a broadband wavelength range.
 44. The hollow-corefibre according to claim 43, wherein the core region and the AREs areconfigured to provide the phase matching of the higher order core modesand the ARE modes in a wavelength range covering up to all wavelengthswithin a hollow-core fibre transparency window of the fundamental coremode.
 45. The hollow-core fibre according to claim 43, wherein the coreregion and the AREs are configured to provide the phase matching of thehigher order core modes and the ARE modes in a wavelength range coveringat least 10 THz.
 46. An optical device, comprising at least onehollow-core fibre of non-bandgap type, the at least one hollow-corefibre including: a hollow core region axially extending along thehollow-core fibre and having a smallest transverse core dimension (D),wherein the core region is configured for guiding a transversefundamental core mode and a plurality of transverse higher order coremodes, and an inner cladding region including an arrangement ofanti-resonant elements (AREs) surrounding the core region along thehollow-core fibre, each having a smallest transverse ARE dimension(d_(i)) and being configured for guiding a plurality of transverse AREmodes, wherein the core region and the AREs are configured to providephase matching of the higher order core modes and the ARE modes, and theARE dimension (d_(i)) and the core dimension (D) are selected such thata ratio of the ARE and core dimensions (d_(i)/D) is approximated to aquotient of zeros of Bessel functions of first kind(u_(lm,ARE)/u_(lm,core)), multiplied with a fitting factor in a rangefrom 0.9 to 1.5, with m being the m-th zero of the Bessel functions offirst kind of order l, said zeros of the Bessel functions describing theLP_(lm) ARE modes and LP_(lm) higher order core modes, respectively. 47.The optical device according to claim 46, comprising at least one of amodal filtering device, a light source, an optical amplifier, a beamdelivery system, a pulse shaper, a data communication system, and afrequency converter.
 48. The optical device according to claim 47,including at least one of the features: the light source is a laser, thepulse shaper is configured for pulse compression, and the frequencyconverter is configured for supercontinuum generation.
 49. A method ofmanufacturing a hollow-core fibre of non-bandgap type, comprising:providing a hollow core region axially extending along the hollow-corefibre and having a smallest transverse core dimension (D), wherein thecore region is configured for guiding a transverse fundamental core modeand transverse higher order core modes, and providing an inner claddingregion comprising an arrangement of anti-resonant elements (AREs)surrounding the core region along the hollow-core fibre, each having asmallest transverse ARE dimension (d_(i)) and being configured forguiding transverse ARE modes, wherein the core region and the AREs areconfigured to provide phase matching of the higher order core modes andthe ARE modes, wherein the ARE dimension (d_(i)) and the core dimension(D) are selected such that a ratio of the ARE and core dimensions(d_(i)/D) is approximated to a quotient of zeros of Bessel functions offirst kind (u_(lm,ARE)/u_(lm,core)), multiplied with a fitting factor ina range from 0.9 to 1.5, with m being the m-th zero of the Besselfunctions of first kind of order l, said zeros of the Bessel functionsdescribing the LP_(lm) ARE modes and LP_(lm) higher order core modes,respectively.
 50. The method according to claim 49, wherein thehollow-core fibre is manufactured with the features of the hollow-corefibre of non-bandgap type, the hollow-core fibre including: a hollowcore region axially extending along the hollow-core fibre and having asmallest transverse core dimension (D), wherein the core region isconfigured for guiding a transverse fundamental core mode and aplurality of transverse higher order core modes, and an inner claddingregion including an arrangement of anti-resonant elements (AREs)surrounding the core region along the hollow-core fibre, each having asmallest transverse ARE dimension (d_(i)) and being configured forguiding a plurality of transverse ARE modes, wherein the core region andthe AREs are configured to provide phase matching of the higher ordercore modes and the ARE modes, and the ARE dimension (d_(i)) and the coredimension (D) are selected such that a ratio of the ARE and coredimensions (d_(i)/D) is approximated to a quotient of zeros of Besselfunctions of first kind (u_(lm,ARE)/u_(lm,core)), multiplied with afitting factor in a range from 0.9 to 1.5, with m being the m-th zero ofthe Bessel functions of first kind of order l, said zeros of the Besselfunctions describing the LP_(lm) ARE modes and LP_(lm) higher order coremodes, respectively.
 51. The method according to claim 49, wherein theARE dimension (d_(i)) is selected by applying an analytical model inwhich the core and the AREs are treated as capillaries, wherein modalindices of the LP_(lm) modes in the capillaries are approximated by$n_{lm} = \sqrt{1 - {( \frac{u_{lm}}{\pi \; f_{s}} )^{2}( \frac{\lambda}{d_{l}} )^{2}}}$wherein u_(lm) is the m-th zero of the Bessel function J_(l), d_(i) isthe inner diameter of the capillary, and the parameter f_(s) is aheuristic fit parameter.
 52. The method according to 49, furthercomprising: (a) providing ARE preforms and a hollow jacket preform, (b)fixing the ARE preforms on an inner surface of the jacket preform in adistributed manner, and (c) heating and drawing the jacket preformincluding the ARE preforms until the ARE and core dimensions are set.53. The method according to claim 52, wherein step (c) includes (c1)heating and drawing the jacket preform including the ARE preforms to acane, and (c2) heating and drawing the cane until the ARE and coredimensions are set.
 54. The method according to claim 53, wherein step(c) includes applying one of a vacuum and an increased pressure to atleast one of the jacket preform, the ARE preforms, and hollow regions ofthe cane for setting the ARE and core dimensions.
 55. The methodaccording to claim 53, further comprising a post-processing step offilling at least one of the core region and the AREs with at least oneof a gas, a liquid, and a material having a non-linear optical response.